Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article
     

On the correspondence of representations between $GL(n)$ and division algebras

Author(s): Joshua Lansky; A. Raghuram
Journal: Proc. Amer. Math. Soc. 131 (2003), 1641-1648.
MSC (2000): Primary 22E35, 22E50
Posted: December 6, 2002
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: For a division algebra $D$ over a $p$-adic field $F,$ we prove that depth is preserved under the correspondence of discrete series representations of $GL_n(F)$ and irreducible representations of $D^*$ by proving that an explicit relation holds between depth and conductor for all such representations. We also show that this relation holds for many (possibly all) discrete series representations of $GL_2(D).$

References:

1.
C.J. Bushnell, Hereditary orders, Gauss sums and supercuspidal representations of ${\rm GL}_N$, J. Reine Angew. Math., 375/376, 184-210 (1987). MR 88e:22024

2.
C.J. Bushnell and J. Fröhlich, Nonabelian congruence Gauss sums and $p$-adic simple algebras, Proc. London Math. Soc., No. 50, 207-264 (1985). MR 86g:11071

3.
C.J. Bushnell and G. Henniart, Local tame lifting for ${\rm GL}(N)$. I. Simple characters, Publ. Math. I.H.E.S., No. 83, 105-233 (1996). MR 98m:11129

4.
C.J. Bushnell and G. Henniart, Local tame lifting for ${\rm GL}(n)$. II. Wildly ramified supercuspidals, Asterisque No. 254, (1999). MR 2000d:11147

5.
P. Deligne, D. Kazhdan and M.-F. Vignéras, Représentations des algèbres centrales simples $p$-adiques, Représentations des Groupes Réductifs sur un Corp Locaux, Hermann, Paris, 33-118 (1984). MR 86h:11044

6.
R. Godement and H. Jacquet, Zeta functions of simple algebras, LNM 260, Springer Verlag, (1972). MR 49:7241

7.
G. Henniart, On the local Langlands conjecture for $GL(n)$ : The cyclic case, Ann. of Math., 123, 143-203 (1986). MR 87k:11132

8.
G. Henniart, Une preuve simple des conjectures de Langlands pour ${\rm GL}(n)$ sur un corps $p$-adique. (French) [A simple proof of the Langlands conjectures for ${\rm GL}(n)$ over a $p$-adic field], Invent. Math., 139, no. 2, 439-455 (2000). MR 2001e:11052

9.
H. Jacquet, I. Pitateskii-Shapiro, J. Shalika, Conducteur des représentations du groupe linéaire. (French) [Conductor of linear group representations], Math. Ann., 256, no. 2, 199-214 (1981). MR 83c:22025

10.
H. Koch and E. W. Zink, Zur korrespondenz von darstellungen der Galoisgruppen und der zentralen divsions algebren uber lokalen korper, Math. Nachr. 98, 83-119 (1980). MR 83h:12025

11.
Stephen S. Kudla, The local Langlands correspondence : The non-Archimedean case, Proc. Symp. in Pure Math., vol. 55, part II, 365-391 (1994). MR 95d:11065

12.
A. Moy and G. Prasad, Unrefined minimal $K$-types for $p$-adic groups, Invent. Math., 116, 393-408 (1994). MR 95f:22023

13.
A. Moy and G. Prasad, Jacquet functors and unrefined minimal $K$-types, Comment. Math. Helevetici, 71, 98-121 (1996). MR 97c:22021

14.
D. Prasad and A. Raghuram, Kirillov theory for $GL_2(\mathcal{D})$ where $\mathcal{D}$ is a division algebra over a non-Archimedean local field, Duke Math. J., Vol. 104, No. 1, 19-44 (2000). MR 2001i:22024

15.
A. Raghuram, Some topics in Algebraic groups : Representation theory of $GL_2(\mathcal{D})$ where $\mathcal{D}$ is a division algebra over a non-Archimedean local field, Thesis, Tata Institute of Fundamental Research, University of Mumbai, (1999).

16.
J. Rogawski, Representations of ${\rm GL}(n)$ and division algebras over a $p$-adic field, Duke Math. J., Vol. 50, No. 1, 161-196 (1983). MR 84j:12018

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 22E35, 22E50

Retrieve articles in all Journals with MSC (2000): 22E35, 22E50

Additional Information:

Joshua Lansky
Affiliation: Department of Mathematics, 380 Olin Science Building, Bucknell University, Lewisburg, Pennsylvania 17837
Email: jlansky@bucknell.edu

A. Raghuram
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Dr. Homi Bhabha Road, Colaba, Mumbai - 400005, India
Email: raghuram@math.tifr.res.in

DOI: 10.1090/S0002-9939-02-06918-6
PII: S 0002-9939(02)06918-6
Received by editor(s): December 19, 2001
Posted: December 6, 2002
Communicated by: Rebecca Herb
Copyright of article: Copyright 2002, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google